Optimal. Leaf size=100 \[ \frac {3 b \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}+\frac {3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 f}-\frac {\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f} \]
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Rubi [A] time = 0.10, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3663, 277, 195, 217, 206} \[ \frac {3 b \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}+\frac {3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 f}-\frac {\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 277
Rule 3663
Rubi steps
\begin {align*} \int \csc ^2(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f}+\frac {(3 b) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {3 b \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}-\frac {\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f}+\frac {(3 a b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac {3 b \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}-\frac {\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f}+\frac {(3 a b) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 f}\\ &=\frac {3 a \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \tan (e+f x)}{\sqrt {a+b \tan ^2(e+f x)}}\right )}{2 f}+\frac {3 b \tan (e+f x) \sqrt {a+b \tan ^2(e+f x)}}{2 f}-\frac {\cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{f}\\ \end {align*}
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Mathematica [C] time = 2.68, size = 220, normalized size = 2.20 \[ \frac {\csc (e+f x) \sec ^3(e+f x) \left (-4 \left (2 a^2+b^2\right ) \cos (2 (e+f x))-2 a^2 \cos (4 (e+f x))-6 a^2+a b \cos (4 (e+f x))+3 \sqrt {2} a b \sin ^2(2 (e+f x)) \sqrt {\frac {\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {\frac {(a+b+(a-b) \cos (2 (e+f x))) \csc ^2(e+f x)}{b}}}{\sqrt {2}}\right )\right |1\right )-a b+b^2 \cos (4 (e+f x))+3 b^2\right )}{8 \sqrt {2} f \sqrt {\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 387, normalized size = 3.87 \[ \left [\frac {3 \, a \sqrt {b} \cos \left (f x + e\right ) \log \left (\frac {{\left (a^{2} - 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} + 8 \, {\left (a b - 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {b} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}} \sin \left (f x + e\right ) + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right ) \sin \left (f x + e\right ) - 4 \, {\left ({\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{8 \, f \cos \left (f x + e\right ) \sin \left (f x + e\right )}, -\frac {3 \, a \sqrt {-b} \arctan \left (\frac {{\left ({\left (a - 2 \, b\right )} \cos \left (f x + e\right )^{3} + 2 \, b \cos \left (f x + e\right )\right )} \sqrt {-b} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{2 \, {\left ({\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 2 \, {\left ({\left (2 \, a + b\right )} \cos \left (f x + e\right )^{2} - b\right )} \sqrt {\frac {{\left (a - b\right )} \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \, f \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \csc \left (f x + e\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.99, size = 1355, normalized size = 13.55 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.67, size = 73, normalized size = 0.73 \[ \frac {3 \, a \sqrt {b} \operatorname {arsinh}\left (\frac {b \tan \left (f x + e\right )}{\sqrt {a b}}\right ) + 3 \, \sqrt {b \tan \left (f x + e\right )^{2} + a} b \tan \left (f x + e\right ) - \frac {2 \, {\left (b \tan \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}}}{\tan \left (f x + e\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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